Why do neural networks need so many training examples to perform? A human child at age 2 needs around 5 instances of a car to be able to identify it with reasonable accuracy regardless of color, make, etc. When my son was 2, he was able to identify trams and trains, even though he had seen just a few. Since he was usually confusing one with each other, apparently his neural network was not trained enough, but still.
What is it that artificial neural networks are missing that prevent them from being able to learn way quicker? Is transfer learning an answer?
 A: 
A human child at age 2 needs around 5 instances of a car to be able to identify it with reasonable accuracy regardless of color, make, etc. 

The concept of "instances" gets easily muddied. While a child may have seen 5 unique instances of a car, they have actually seen thousands of thousands of frames, in many differing environments. They have likely seen cars in other contexts. They also have an intuition for the physical world developed over their lifetime - some transfer learning probably happens here. Yet we wrap all of that up into "5 instances."
Meanwhile, every single frame/image you pass to a CNN is considered an "example." If you apply a consistent definition, both systems are really utilizing a much more similar amount of training data. 
Also, I would like to note that convolutional neural networks - CNNs - are more useful in computer vision than ANNs, and in fact approach human performance in tasks like image classification. Deep learning is (probably) not a panacea, but it does perform admirably in this domain.
A: As pointed out by others, the data-efficiency of artificial neural networks varies quite substantially, depending on the details. As a matter of fact, there are many so called one-shot learning methods, that can solve the task of labelling trams with quite good accuracy, using only a single labelled sample. 
One way to do this is by so-called transfer learning; a network trained on other labels is usually very effectively adaptable to new labels, since the hard work is breaking down the low level components of the image in a sensible way.
But we do not infact need such labeled data to perform such task; much like babies dont need nearly as much labeled data as the neural networs you are thinking of do.
For instance, one such unsupervised methods that I have also successfully applied in other contexts, is to take an unlabeled set of images, randomly rotate them, and train a network to predict which side of the image is 'up'. Without knowing what the visible objects are, or what they are called, this forces the network to learn a tremendous amount of structure about the images; and this can form an excellent basis for much more data-efficient subsequent labeled learning.
While it is true that artificial networks are quite different from real ones in probably meaningful ways, such as the absence of an obvious analogue of backpropagation, it is very probably true that real neural networks make use of the same tricks, of trying to learn the structure in the data implied by some simple priors. 
One other example which almost certainly plays a role in animals and has also shown great promise in understanding video, is in the assumption that the future should be predictable from the past. Just by starting from that assumption, you can teach a neural network a whole lot. Or on a philosophical level, I am inclined to believe that this assumption underlies almost everything what we consider to be 'knowledge'.
I am not saying anything new here; but it is relatively new in the sense that these possibilities are too young to have found many applications yet, and do not yet have percolated down to the textbook understanding of 'what an ANN can do'. So to answer the OPs question; ANN's have already closed much of the gap that you describe.
A: One thing that I haven't seen in the answers so far is the fact that one 'instance' of a real world object that is seen by a human child does not corresponds to an instance in the context of NN training. 
Suppose you're standing at a railway intersection with a 5 year old child and watch 5 trains pass within 10 minutes. Now, you could say "My child only saw 5 trains and can reliably identify other trains while a NN needs thousands of images!". While this is likely true, you are completely ignoring the fact that every train your child sees contains A LOT more information than a single image of a train. In fact, the brain of your child is processing several dozens images of the train per second while it is passing by, each from a slightly different angle, different shadows, etc., while a single image will provide the NN with very limited information. 
In this context, your child even has information that is not available to the NN, for example the speed of the train or the sound that the train makes. 
Further, your child can talk and ASK QUESTIONS! "Trains are very long, right?" "Yes.", "And they are very big too, right?" "Yes.". With two simple questions your child learn two very essential features in less than a minute!
Another important point is object detection. Your child is able to identify immediately on which object, i.e. which part of the image, it needs to focus on, while a NN must learn to detect the relevant object before it can attempt to classify it. 
A: First of all, at age two, a child knows a lot about the world and actively applies this knowledge. A child does a lot of "transfer learning" by applying this knowledge to new concepts.
Second, before seeing those five "labeled" examples of cars, a child sees a lot of cars on the street, on TV, toy cars, etc., so also a lot of "unsupervised learning" happens beforehand.
Finally, neural networks have almost nothing in common with the human brain, so there's not much point in comparing them. Also notice that there are algorithms for one-shot learning, and pretty much research on it currently happens.
A: One way to train a deep neural network is to treat it as a stack of auto-encoders (Restricted Boltzmann Machines).
In theory, an auto-encoder learns in an unsupervised manner:  It takes arbitrary, unlabelled input data and processes it to generate output data.  Then it takes that output data, and tries to regenerate its input data.  It tweaks its nodes' parameters until it can come close to round-tripping its data.  If you think about it, the auto-encoder is writing its own automated unit tests.  In effect, it is turning its "unlabelled input data" into labelled data:  The original data serves as a label for the round-tripped data.
After the layers of auto-encoders are trained, the neural network is fine-tuned using labelled data to perform its intended function.  In effect, these are functional tests.
The original poster asks why a lot of data is needed to train an artificial neural network, and compares that to the allegedly low amount of training data needed by a two-year-old human.  The original poster is comparing apples-to-oranges:  The overall training process for the artificial neural net, versus the fine-tuning with labels for the two-year-old.
But in reality, the two-year old has been training its auto-encoders on random, self-labelled data for more than two years.  Babies dream when they are in utero.  (So do kittens.)  Researchers have described these dreams as involving random neuron firings in the visual processing centers.
A: We don't learn to "see cars" until we learn to see
It takes quite a long time and lots of examples for a child to learn how to see objects as such. After that, a child can learn to identify a particular type of object from just a few examples. If you compare a two year old child with a learning system that literally starts from a blank slate, it's an apples and oranges comparison; at that age child has seen thousands of hours of "video footage".
In a similar manner, it takes artificial neural networks a lot of examples to learn "how to see" but after that it's possible to transfer that knowledge to new examples. Transfer learning is a whole domain of machine learning, and things like "one shot learning" are possible - you can build ANNs that will learn to identify new types of objects that it hasn't seen before from a single example, or to identify a particular person from a single photo of their face. But doing this initial "learning to see" part well requires quite a lot of data.
Furthermore, there's some evidence that not all training data is equal, namely, that data which you "choose" while learning is more effective than data that's simply provided to you. E.g. Held & Hein twin kitten experiment. https://www.lri.fr/~mbl/ENS/FONDIHM/2013/papers/about-HeldHein63.pdf 
A: One major aspect that I don't see in current answers is evolution.
A child's brain does not learn from scratch. It's similar to asking how deer and giraffe babies can walk a few minutes after birth. Because they are born with their brains already wired for this task. There is some fine-tuning needed of course, but the baby deer doesn't learn to walk from "random initialization".
Similarly, the fact that big moving objects exist and are important to keep track of is something we are born with.
So I think the presupposition of this question is simply false. Human neural networks had the opportunity to see tons of - maybe not cars but - moving, rotating 3D objects with difficult textures and shapes etc., but this happened through lots of generations and the learning took place by evolutionary algorithms, i.e. the ones whose brain was better structured for this task, could live to reproduce with higher chance, leaving the next generation with better and better brain wiring from the start.
A: I would argue the performance is not that different as you might expect, but you ask a great question (see the last paragraph).
As you mention transfer learning: To compare apples with apples we have to look how many pictures in total and how many pictures of the class of interest a human / neural net "sees".
1. How many pictures does a human look at?
Human´s eye movement takes around 200ms which could be seen as kind of an "biological photo". See the talk by computer vision expert Fei-Fei Li: https://www.ted.com/talks/fei_fei_li_how_we_re_teaching_computers_to_understand_pictures#t-362785. 
She adds:

So by age 3 a child would have seen hundreds of millions of pictures.

In ImageNet, the leading database for object detection, there are ~14million labeled pictures. So a neural network being trained on ImageNet would have seen as many pictures as a 14000000/5/60/60/24*2 ~ 64 days old baby, so two months old (assuming the baby is awake half of her life). 
To be fair its hard to tell how many of this pictures are labeled. Moreover, the pictures, a baby sees, are not that diverse like in ImageNet. (Probably the baby sees her mother have of the time,... ;). 
However, i think its fair to say that your son will have seen hundreds of millions of pictures (and then applies transfer learning).
So how many pictures do we need to learn a new category given a solid base of related pictures that can be (transfer) learned from?
First blog post i found was this: https://blog.keras.io/building-powerful-image-classification-models-using-very-little-data.html. They use 1000 examples per class. I could imagine 2.5 years later even way less is required.
However, 1000 pictures can be seen by a human in 1000/5/60 in 3.3 minutes.
You wrote:

A human child at age 2 needs around 5 instances of a car to be able to
  identify it with reasonable accuracy regardless of color, make, etc.

That would be equivilant to forty seconds per instance (with various angles of that object to make it comparable). 
To sum up:
As i mentioned, I had to make a few assumptions. But i think, one can see that the performance is not that different as one might expect.
However, i believe you ask a great question and here is why:
2. Would neural network perform better/different if they would work more like brains? (Geoffrey Hinton says yes).
In an interview https://www.wired.com/story/googles-ai-guru-computers-think-more-like-brains/, in late 2018, he compares the current implementations of neural networks with the brain. He mentions, in terms of weights, the artificial neural networks are smaller than the brain by a factor of 10.000. Therefore, the brain needs way less iterations of trainings to learn. In order to enable artificial neural networks, to work more like our brains, he follows another trend in hardware, a UK based startup called Graphcore. It reduces the calculation time by a smart way of storing the weights of a neural network. Therefore, more weights can be used and the training time of the artificial neural networks might get reduced. 
A: I don't know much about neural networks but I know a fair bit about babies.
Many 2 year olds have a lot of issues with how general words should be. For instance, it is quite common at that age for kids to use "dog" for any four legged animal. That's a more difficult distinction than "car" - just think how different a poodle looks from a great Dane, for instance and yet they are both "dog" while a cat is not.
And a child at 2 has seen many many more than 5 examples of "car". A kid sees dozens or even hundreds of examples of cars any time the family goes for a drive. And a lot of parents will comment "look at the car" a lot more than 5 times. But kids can also think in ways that they weren't told about. For instance, on the street the kid sees lots of things lined up. His dad says (of one) "look at the shiny car!" and the kid thinks "maybe all those other things lined up are also cars?"
A: I am an expert in this. I am human, I was a baby, I have a car, and I do AI. 
The reason why babies pick up cars with far more limited examples is intuition. The human brain already has structures to deal with 3D rotations. Also, there are two eyes which provide parallax for depth mapping which really helps. You can intuit between a car and a picture of a car, because there is no actual depth to the picture. Hinton (AI researcher) has proposed the idea of Capsule Networks, which would be able to handle things more intuitively. Unfortunately for computers, the training data is (usually) 2D images, arrays of flat pixels. In order to not over-fit, much data is required so the orientation of the cars in the images is generalized. The baby brain can do this already and can recognize a car at any orientation. 
A: I caution against expecting strong resemblance between biological and artificial neural networks. I think the name "neural networks" is a bit dangerous, because it tricks people into expecting that neurological processes and machine learning should be the same. The differences between biological and artificial neural networks outweigh the similarities.
As an example of how this can go awry, you can also turn the reasoning in the original post on its head. You can train a neural network to learn to recognize cars in an afternoon, provided you have a reasonably fast computer and some amount of training data. You can make this a binary task (car/not car) or a multi-class task (car/tram/bike/airplane/boat) and still be confident in a high level of success. 
By contrast, I wouldn't expect a child to be able to pick out a car the day - or even the week - after it's born, even after it has seen "so many training examples." Something is obviously different between a two-year-old and an infant that accounts for the difference in learning ability, whereas a vanilla image classification neural network is perfectly capable of picking up object classification immediately after "birth." I think that there are two important differences: (1) the relative volumes of training data available and (2) a self-teaching mechanism that develops over time because of abundant training data.

The original post exposes two questions. The title and body of the question ask why neural networks need "so many examples." Relative to a child's experience, neural networks trained using common image benchmarks have comparatively little data.
I will re-phrases the question in the title to 
"How does training a neural network for a common image benchmark compare & contrast to the learning experience of a child?"
For the sake of comparison I'll consider the CIFAR-10 data because it is a common image benchmark. The labeled portion is composed of 10 classes of images with 6000 images per class. Each image is 32x32 pixels. If you somehow stacked the labeled images from CIFAR-10 and made a standard 48 fps video, you'd have about 20 minutes of footage.
A child of 2 years who observes the world for 12 hours daily has roughly 263000 minutes (more than 4000 hours) of direct observations of the world, including feedback from adults (labels). (These are just ballpark figures -- I don't know how many minutes a typical two-year-old has spent observing the world.) Moreover, the child will have exposure to many, many objects beyond the 10 classes that comprise CIFAR-10.
So there are a few things at play. One is that the child has exposure to more data overall and a more diverse source of data than the CIFAR-10 model has. Data diversity and data volume are well-recognized as pre-requisites for robust models in general. In this light, it doesn't seem surprising that a neural network is worse at this task than the child, because a neural network trained on CIFAR-10 is positively starved for training data compared to the two-year-old. The image resolution available to a child is better than the 32x32 CIFAR-10 images, so the child is able to learn information about the fine details of objects.
The CIFAR-10 to two-year-old comparison is not perfect because the CIFAR-10 model will likely be trained with multiple passes over the same static images, while the child will see, using binocular vision, how objects are arranged in a three-dimensional world while moving about and with different lighting conditions and perspectives on the same objects.
The anecdote about OP's child implies a second question, 
"How can neural networks become self-teaching?"
A child is endowed with some talent for self-teaching, so that new categories of objects can be added over time without having to start over from scratch. 


*

*OP's remark about transfer-learning names one kind of model adaptation in the machine learning context.

*In comments, other users have pointed out that one- and few-shot learning* is another machine learning research area.

*Additionally, reinforcement-learning addresses self-teaching models from a different perspective, essentially allowing robots to undertake trial-and-error experimentation to find optimal strategies for solving specific problems (e.g. playing chess).
It's probably true that all three of these machine learning paradigms are germane to improving how machines adapt to new computer vision tasks. Quickly adapting machine learning models to new tasks is an active area of research. However, because the practical goals of these projects (identify new instances of malware, recognize imposters in passport photos, index the internet) and criteria for success differ from the goals of a child learning about the world, and the fact that one is done in a computer using math and the other is done in organic material using chemistry, direct comparisons between the two will remain fraught.

As an aside, it would be interesting to study how to flip the CIFAR-10 problem around and train a neural network to recognize 6000 objects from 10 examples of each. But even this wouldn't be a fair comparison to 2-year-old, because there would still be a large discrepancy in the total volume, diversity and resolution of the training data.
*We don't presently have a tags for one-shot learning or few-shot learning.
A: This is an a fascinating question that I've pondered over a lot also, and can come up with a few explanations why.


*

*Neural networks work nothing like the brain. Backpropagation is unique to neural networks, and does not happen in the brain. In that sense, we just don't know the general learning algorithm in our brains. It could be electrical, it could be chemical, it could even be a combination of the two. Neural networks could be considered an inferior form of learning compared to our brains because of how simplified they are.

*If neural networks are indeed like our brain, then human babies undergo extensive "training" of the early layers, like feature extraction, in their early days. So their neural networks aren't really trained from scratch, but rather the last layer is retrained to add more and more classes and labels.

